Step by step
The question
Question: simplify 3 over √2.
3√2
1 Remove the root at the bottom
A root times itself is the number inside: √2 times √2 is 2.
So we multiply the bottom by √2.
√2 × √2 = 2
To keep the fraction the same, we multiply the top by √2 too.
√2 over √2 is 1, and times 1 changes nothing.
√2√2 = 1
So we multiply by √2 over √2.
= 3√2 × √2√2
Careful!
Many people multiply only the bottom.
Then the fraction changes.
Top and bottom get the same √2.
Careful: multiply the TOP by the root too
On top: 3 times √2 is 3√2.
On the bottom: √2 times √2 is 2.
No root at the bottom now.
The numbers can't be cancelled, so we are done.
= 3√22
2 Check
Let's check with a calculator.
The start is about 2.12.
Start: 3√2 ≈ 2.12
The answer is about 2.12 too.
So it is right!
Answer: 3√22 ≈ 2.12
Let's look back.
First: remove the root at the bottom.
Last: check.
Answer
= 3√22